To solve this triangle, let’s apply the Law of Sines. Given:

• $a = 23.07$ cm
• $b = 10.75$ cm
• $A = 30.2^\circ$

Since we have two sides and one angle (SAA or SSA), let's proceed as follows:

Step 1: Find Angle $B$ Using the Law of Sines

The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B}$

Rearrange to solve for $B$: $\sin B = \frac{b \cdot \sin A}{a}$

Plugging in the values: $\sin B = \frac{10.75 \cdot \sin(30.2^\circ)}{23.07}$

Calculating $\sin(30.2^\circ) \approx 0.5026$: $\sin B = \frac{10.75 \cdot 0.5026}{23.07} \approx 0.2343$

Now, $B = \sin^{-1}(0.2343) \approx 13.6^\circ$.

Step 2: Determine if There’s a Second Solution

Since $\sin B \approx 0.2343$, there’s a possibility of two triangles (ambiguous case). The second possible value of $B$ could be: $B' = 180^\circ - 13.6^\circ = 166.4^\circ$

However, if $B = 166.4^\circ$, then $A + B = 30.2^\circ + 166.4^\circ = 196.6^\circ$, which is impossible in a triangle (angles must sum up to $180^\circ$). So, there’s only one solution for $B \approx 13.6^\circ$.

Step 3: Find Angle $C$

Using the angle sum in a triangle: $C = 180^\circ - A - B$ $C = 180^\circ - 30.2^\circ - 13.6^\circ \approx 136.2^\circ$

Step 4: Find Side $c$ Using the Law of Sines

Now, use the Law of Sines again to find $c$: $\frac{c}{\sin C} = \frac{a}{\sin A}$ $c = \frac{a \cdot \sin C}{\sin A}$

Calculating $\sin(136.2^\circ) \approx 0.7314$: $c = \frac{23.07 \cdot 0.7314}{0.5026} \approx 33.56 \text{ cm}$

Summary of Solution

• Angle $B \approx 13.6^\circ$
• Angle $C \approx 136.2^\circ$
• Side $c \approx 33.56$ cm

Answer Choice

The correct answer is:

• B. There is 1 possible solution to the triangle.

Would you like more details on any of these steps or have further questions?

Related Questions

1. How is the Law of Sines used in ambiguous cases?
2. Why does the SSA configuration sometimes lead to two possible triangles?
3. What other methods can be used to solve triangles when given different sets of information?
4. How can the Law of Cosines be applied if all sides are known?
5. What happens if the angle sum exceeds 180° in a triangle?

Tip

In ambiguous cases (SSA), always check if a second triangle is possible by testing if $180^\circ - B$ can also satisfy the triangle’s angle sum.